<ul><li> 1. P r e f The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used. Every effort has been made to present results concisely as well as precisely SOthat they may be referred to with a maxi- mum of ease as well as confidence. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance. This wide coverage of topics has been adopted SOas to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment. The book is divided into two main parts. Part 1 presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker. Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are sep- arated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SOthat there is no need to be concerned about the possibility of errer due to looking in the wrong column or row. 1 wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook. Appropriate references to such sources are given next to the corresponding tables. In particular 1 am indebted to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S t a t i s tT a b l e sf o yB i o l oA g r i ca n dM e d iR e s 1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation. M. R. SPIEGEL Rensselaer Polytechnic Institute September, 1968 </li></ul><p> 2. CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 2s. 26. 27. 28. 29. 30. Page Special Constants. ............................... 1 Special Products and Factors .......................... 2 The Binomial Formula and Binomial Coefficients ................. 3 Geometric Formulas .............................. 5 Trigonometric Functions ............................ 11 Complex Numbers ................................ 21 Exponential and Logarithmic Functions ..................... 23 Hyperbolic Functions .............................. 26 Solutions of Algebraic Equations ........................ 32 Formulas from Plane Analytic Geometry .................... 34 Special Plane Curves....~ .......................... 40 Formulas from Solid Analytic Geometry .................... 46 Derivatives ................................... 53 Indefinite Integrals ............................... 57 Definite Integrals ................................ 94 The Gamma Function ............................. .10 1 The Beta Function .............................. .lO3 Basic Differential Equations and Solutions ................... .104 Series of Constants...............................lO 7 Taylor Series..................................ll 0 Bernoulliand Euler Numbers ......................... .114 Formulas from Vector Analysis. ....................... .116 Fourier Series ................................ .~31 Bessel Functions. .............................. .136 Legendre Functions...............................l4 6 Associated Legendre Functions ......................... .149 Hermite Polynomials..............................l5 1 Laguerre Polynomials ............................. .153 Associated Laguerre Polynomials ........................ KG Chebyshev Polynomials.............................l5 7 3. Part I FORMULAS 4. Greek name Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda MU THE GREEK ALPHABET G&amp;W A B l? A E Z H (3 1 K A M Greek name Nu Xi Omicron Pi Rho Sigma Tau Upsilon Phi Chi Psi Omega Greek Lower case tter Capital N sz 0 IT P 2 T k @ X * n 5. 1.1 1.2 = natural base of logarithms 1.3 fi = 1.41421 35623 73095 04889. 1.4 fi = 1.73205 08075 68877 2935. . . 1.5 fi = 2.23606 79774 99789 6964. . 1.6 h = 1.25992 1050. . 1.7 &amp; = 1.44224 9570. . 1.8 fi = 1.14869 8355. . 1.9 b = 1.24573 0940. . 1.10 eT = 23.14069 26327 79269 006. . 1.11 re = 22.45915 77183 61045 47342 715. . 1.12 ee = 15.15426 22414 79264 190. . 1.13 logI, 2 = 0.30102 99956 63981 19521 37389. . 1.14 logI, 3 = 0.47712 12547 19662 43729 50279. . 1.15 logIO e = 0.43429 44819 03251 82765. . 1.16 logul ?r = 0.49714 98726 94133 85435 12683. . . 1.17 loge 10 = In 10 = 2.30258 50929 94045 68401 7991. . 1.18 loge 2 = ln 2 = 0.69314 71805 59945 30941 7232. . 1.19 loge 3 = ln 3 = 1.09861 22886 68109 69139 5245. . 1.20 y = 0.57721 56649 01532 86060 6512. . = Eukr's co%stu~t 1.21 ey = 1.78107 24179 90197 9852. . [see 1.201 1.22 fi = 1.64872 12707 00128 1468. . 1.23 6 = r(&amp;) = 1.77245 38509 05516 02729 8167. . where F is the gummu ~ZLYLC~~OTZ[sec pages 101-102). 1.24 1.25 1-26 1.27 II(&amp;) = 2.67893 85347 07748. . r(i) = 3.62560 99082 21908. . 1 radian = 180/7r = 57.29577 95130 8232. .O 1 = ~/180 radians = 0.01745 32925 19943 29576 92. . . radians 1 6. 4 THE BINOMIAL FORMULA AND BINOMIAL COElFI?ICIFJNTS PROPERTIES OF BINOMIAL COEFFiClEblTS 3.6 This leads to Paseals triangk [sec page 2361. 3.7 (1) + (y) + (;) + .. + (1) = 27l 3.8 (1) - (y) + (;) - .+-w(;) = 0 3.9 3.10 (;) + (;) + (7) + .*. = 2n-1 3.11 (y) + (;) + (i) + .* = 2n-1 3.12 3.13 -d 3.14 q+n2+ .. +np = 72. MUlTlNOMlAk FORfvlUlA 3.16 (zI+%~+..+zp)~ = ~~~!~~~~~.~~!~~1~~2..~~~ where the mm, denoted by 2, is taken over a11 nonnegative integers % %, .,np fox- whkh 7. 1 4 GEUMElRlC FORMULAS &amp; RECTANGLE OF LENGTH b AND WIDTH a 4.1 Area = ab 4.2 Perimeter = 2a + 2b b Fig. 4-1 PARAllELOGRAM OF ALTITUDE h AND BASE b 4.3 Area = bh = ab sin e 4.4 Perimeter = 2a + 2b 1 Fig. 4-2 fRlAMf3i.E OF ALTITUDE h AND BASE b 4.5 Area = +bh = +ab sine ZZZI/S(S - a)(s - b)(s - c) where s = &amp;(a+ b + c) = semiperimeter * b 4.6 Perimeter = u + b + c Fig. 4-3 L,Z n_ ., : fRAPB%XD C?FAt.TlTUDE fz AND PARAl.lEL SlDES u AND b., 4.7 Area = 3h(a + b) 4.8 Perimeter = a + b + h C Y&amp;+2 sin 4 = a + b + h(csc e + csc $) /c- 1 Fig. 4-4 5 / - 8. 6 GEOMETRIC FORMULAS REGUkAR POLYGON OF n SIDES EACH CJf 1ENGTH b 4.9 COS(AL) Area = $nb?- cet c = inbz- sin (~4%) 4.10 Perimeter = nb Fig. 4-5 CIRLE OF RADIUS r 4.11 Area = &amp; 7, 00.4.12 Perimeter = 277r Fig. 4-6 SEClOR OF CIRCLE OF RAD+US Y 4.13 Area = &amp;r% [e in radians] T A 8 4.14 Arc length s = ~6 0 T Fig. 4-7 RADIUS OF C1RCJ.E INSCRWED tN A TRtANGlE OF SIDES a,b,c * 4.15 r= &amp;$.s- U)(SY b)(s -.q) s where s = +(u + b + c) = semiperimeter Fig. 4-6 RADIUS- OF CtRClE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c 4.16 R= abc 4ds(s - a)@ - b)(s - c) where e = -&amp;(a.+ b + c) = semiperimeter Fig. 4-9 9. G E O M E TF O R M U7 4 . 1 7A r e a= &amp; n r 2s i ns = 3 6 0 + n r 2s i nn 4 . 1 8P e r i m e t e= 2 n rs i n z= 2 n rs i ny Fig. 4-10 4 . 1 9A r e a= n r 2t a nZ T= n r 2t a nL ! T ! ! ? n n I T 4 . 2 0P e r i m e t e= 2 n rt a n k= 2 n rt a n ? 0 : F i g4 - 1 SRdMMHW W C%Ct&amp; OF RADWS T 4 . 2 1A r e ao fs h a d e dp a r t= + r 2( e- s i ne) e T r tz!? Fig. 4-12 4 . 2 2 4 . 2 3 A r e a= r a b 5 7r/2 P e r i m e t e= 4a 4 1 - kz s i +e c l @ 0 = 27r@sTq [ a p p r o w h e r ek = ~/=/a. See p a g e254 f o rn u m e r it a b l eF i g4 - 1 4 . 2 4A r e a= $ab 4 . 2 5A r cl e n g t hABC = -&amp;dw + E l n 4 a + @ T 1 ) AOC b Fig. 4-14 f- 10. 8 GEOMETRIC FORMULAS RECTANGULAR PARALLELEPIPED OF LENGTH u, HEIGHT r?,WIDTH c 4.26 Volume = ubc 4.27 Surface area = Z(ab + CLC+ bc) PARALLELEPIPED OF CROSS-SECTIONAL AREA A AND HEIGHT h 4.28 Volume = Ah = abcsine 4.29 4.30 4.31 4.32 4.33 4.34 a Fig. 4-15 Fig. 4-16 SPHERE OF RADIUS ,r Volume = + Surface area = 4wz 1 ,------- ---x . @ Fig. 4-17 RIGHT CIRCULAR CYLINDER OF RADIUS T AND HEIGHT h Volume = 77&amp;2 Lateral surface area = 25dz h Fig. 4-18 CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT 2 Volume = m2h = ~41 sine 2wh Lateral surface area = 2777-1 = z = 2wh csc e Fig. 4-19 11. GEOMETRIC FORMULAS 9 CYLINDER OF CROSS-SECTIONAL AREA A AND SLANT HEIGHT I 4.35 Volume = Ah = Alsine 4.36 Ph -Lateral surface area = pZ = G - ph csc t Note that formulas 4.31 to 4.34 are special cases. Fig. 4-20 RIGHT CIRCULAR CONE OF RADIUS ,r AND HEIGHT h 4.37 Volume = jw2/z 4.38 Lateral surface area = 77rd77-D = ~-7-1 Fig. 4-21 PYRAMID OF BASE AREA A AND HEIGHT h 4.39 Volume = +Ah Fig. 4-22 SPHERICAL CAP OF RADIUS ,r AND HEIGHT h 4.40 Volume (shaded in figure) = &amp;rIt2(3v- h) 4.41 Surface area = 2wh Fig. 4-23 FRUSTRUM OF RIGHT CIRCULAR CONE OF RADII u,h AND HEIGHT h 4.42 Volume = +h(d + ab + b2) 4.43 Lateral surface area = T(U+ b) dF + (b - CL)~ = n(a+b)l Fig. 4-24 12. 10 GEOMETRIC FORMULAS SPHEMCAt hiiWW OF ANG%ES A,&amp;C Ubl SPHERE OF RADIUS Y 4.44 Area of triangle ABC = (A + B + C - z-)+ Fig. 4-25 TOW$ &amp;F lNN8R RADlU5 a AND OUTER RADIUS b 4.45 4.46 Volume = &amp;z-~(u+ b)(b - u)~ w Surface area = 7r2(b2- u2) 4.47 Volume = $abc Fig. 4-27 PARAWlO~D aF REVOllJTlON T. 4.4a Volume = &amp;bza Fig. 4-28 13. 5 TRtGOhiOAMTRiC WNCTIONS D E F l N l TO FT R I G O NF U N C TF O RA R I GT R I Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. The trigonometric functions of angle A are defined as follows. 5 . 1sintzof A = sin A = : = opposite B hypotenuse 5 . 2cosineof A = ~OSA = i = adjacent hypotenuse 5 . 3 5 . 4 5.5 opposite tangent of A = tanA = f = -~ adjacent c o t c z n gof A = cet A = k = adjacent opposite A hypotenuse secant of A = sec A = t = -~ adjacent 5 . 6cosecant of A = csc A = z = hypotenuse opposite Fig. 5-1 E X T E N S IT OA N G L EW H I C HM A Y3 EG R E AT H A9 0 Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates (%,y) where x is eonsidered as positive along OX and negative along OX while y is positive along OY and negative along OY. The distance from origin 0 to point P is positive and denoted by r = dm. The angle A described cozmtwcZockwLse from OX is considered pos&amp;ve. If it is described dockhse from OX it is considered negathe. We cal1 XOX and YOY the x and y axis respectively. The various quadrants are denoted by 1,II, III and IV called the first, second, third and fourth quad- rants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A is in the third quadrant. Y Y II 1 II 1 III IV III IV Y Y Fig. 5-2 Fig. 5-3 11 f 14. 12 TRIGONOMETRIC FUNCTIONS For an angle A in any quadrant the trigonometric functions of A are defined as follows. 5.7 sin A = ylr 5.8 COSA = xl?. 5.9 tan A = ylx 5.10 cet A = xly 5.11 sec A = v-lx 5.12 csc A = riy RELAT!ONSHiP BETWEEN DEGREES AN0 RAnIANS A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r. Since 2~ radians = 360 we have 5.13 1 radian = 180/~ = 57.29577 95130 8232. . . o 5.14 10 = ~/180 radians = 0.01745 32925 19943 29576 92. .radians N 1 r e B 0 r M Fig. 5-4 REkATlONSHlPS AMONG TRtGONOMETRK FUNCTItB4S 5.15 tanA = 5 5.19 sineA + ~OS~A = 1 5.16 1 COSA &amp;A ~II ~ zz - tan A sin A 5.20 sec2A - tane A = 1 5.17 1 sec A = ~ COSA 5.21 csceA - cotsA = 1 5.18 1 cscA = - sin A SIaNS AND VARIATIONS OF TRl@ONOMETRK FUNCTIONS 1 + + + + + + 0 to 1 1 to 0 0 to m CCto 0 1 to uz mto 1 - -II + + 1 to 0 0 to -1 -mtoo oto-m -cc to -1 1 to ca - III + + 0 to -1 -1 to 0 0 to d Ccto 0 -1to-m --COto-1 - IV + - + - -1 to 0 0 to 1 -- too oto-m uzto 1 -1 to -- 15. TRIGONOMETRIC FUNCTIONS 1 3 E X A C TV A L U E SF O RT R I G O NF U N C TO FV A R IA N G Angle A Angle A in degrees in radians sin A COSA tan A cet A sec A csc A 00 0 0 1 0 w 1 cc 15O rIIl2 #-fi) &amp;(&amp;+fi) 2-fi 2+* fi-fi &amp;+fi 300 ii/6 1 +ti *fi fi $fi 2 450 zl4 J-fi $fi 1 1 fi fi 60 VI3 Jti r1 fi .+fi 2 ;G 750 5~112 i(fi+m @-fi) 2+&amp; 2-&amp; &amp;+fi fi-fi 900 z.12 1 0 *CU 0 km 1 105O 7~112 *(fi+&amp;) -&amp;(&amp;-Y% -(2+fi) -(2-&amp;) -(&amp;+fi) fi-fi 120 2~13 *fi -* -fi -$fi -2 ++ 1350 3714 +fi -*fi -1 -1 -fi h 150 5~16 4 -+ti -*fi -fi -+fi 2 165O llrll2 $(fi- fi) -&amp;(G+ fi) -(2-fi) -(2+fi) -(fi-fi) Vz+V-c? 180 ?r 0 -1 0 Tm -1 *ca 1950 13~112 -$(fi-fi) -*(&amp;+fi) 2-fi 2 + ti -(&amp;-fi) -(&amp;+fi) 210 7716 1 - 4 6&amp; l 3f i - g f i-2 225O 5z-14 -Jfi -*fi 1 1 -fi -fi 240 4%J3 -# -4 ti &amp;fi -2 -36 255O 17~112 -&amp;&amp;+&amp;Q -&amp;(&amp;-fi) 2+fi 2-6 -(&amp;+?cz) -(fi-fi) 270 3712 -1 0 km 0 Tm -1 285O 19?rll2 -&amp;(&amp;+fi) *(&amp;-fi) -(2+6) -@-fi) &amp;+fi -(fi-fi) 3000 5rl3 -*fi 2 -ti -*fi 2 -$fi 315O 7?rl4 -4fi *fi -1 -1 fi -fi 330 117rl6 1 *fi -+ti -ti $fi -2 345O 237112 -i(fi- 6) &amp;(&amp;+ fi) -(2 - fi) -(2+6) fi-fi -(&amp;+fi) 360 2r 0 1 0 T-J 1 ?m For tables involving other angles see pages 206-211 and 212-215. f 16. 19 5.89 y = cet-1% 5.90 y = sec-l% 5.91 y = csc-lx Fig. 5-14 Fig. 5-15 Fig. 5-16 TRIGONOMETRIC FUNCTIONS I Y _--/ T / /A-- / , --- -77 -- // , RElAilONSHfPS BETWEEN SIDES AND ANGtGS OY A PkAtM TRlAF4GlG The following results hold for any plane triangle ABC with sides a, b, c and angles A, B, C. 5.92 Law of Sines a b c-=Y=- sin A sin B sin C 5.93 Law of Cosines A 1 /A C f with 5.94 Law with 5.95 cs = a2 + bz - Zab COS C similar relations involving the other sides and angles. of Tangents a+b tan $(A + B) - = tan i(..</p>